| L(s) = 1 | − 8·5-s + 6·9-s + 16·16-s + 24·23-s − 36·25-s + 112·31-s + 80·37-s − 48·45-s − 8·47-s + 192·49-s − 40·53-s − 64·59-s − 160·67-s − 200·71-s − 128·80-s + 27·81-s + 144·89-s − 140·97-s − 76·103-s − 392·113-s − 192·115-s + 552·125-s + 127-s + 131-s + 137-s + 139-s + 96·144-s + ⋯ |
| L(s) = 1 | − 8/5·5-s + 2/3·9-s + 16-s + 1.04·23-s − 1.43·25-s + 3.61·31-s + 2.16·37-s − 1.06·45-s − 0.170·47-s + 3.91·49-s − 0.754·53-s − 1.08·59-s − 2.38·67-s − 2.81·71-s − 8/5·80-s + 1/3·81-s + 1.61·89-s − 1.44·97-s − 0.737·103-s − 3.46·113-s − 1.66·115-s + 4.41·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 2/3·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.006668053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.006668053\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 4 T + 42 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 192 T^{2} + 14015 T^{4} - 192 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 16 p T^{2} + 52386 T^{4} - 16 p^{5} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 820 T^{2} + 307494 T^{4} - 820 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1120 T^{2} + 554559 T^{4} - 1120 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 12 T - 106 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 108 T^{2} + 1403606 T^{4} + 108 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 56 T + 2703 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 40 T + 615 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6100 T^{2} + 14856822 T^{4} - 6100 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 5008 T^{2} + 13022946 T^{4} - 5008 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 4 T - 378 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 20 T + 2646 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 32 T + 6918 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 6528 T^{2} + 37724303 T^{4} - 6528 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 80 T + 8991 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 100 T + 11994 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 12720 T^{2} + 78809759 T^{4} - 12720 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16816 T^{2} + 134402751 T^{4} - 16816 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1908 T^{2} - 14635114 T^{4} - 1908 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 72 T + 16706 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 70 T + 9243 T^{2} + 70 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.016802058409283456141586953644, −7.68187702819058762865372134318, −7.63854646362075158856285490530, −7.41884043844711695175031559377, −7.30798959440475872406608845753, −6.77708597494169768352291693761, −6.55797005893341862109985690471, −6.34499654029659436674064982108, −5.84885051643634919623635173668, −5.78186015180769933607696304184, −5.70963941476319391961838214009, −5.10687973454071378717316876648, −4.61827144021653111063085260929, −4.47819429893413615539785673564, −4.40699590305101401971494272686, −4.03260445489876216248547779420, −3.79220911587343167629823016699, −3.47198356819680165540182221223, −2.90078506726802688972652095096, −2.67885359059608155833061523795, −2.62350987118371064708438611967, −1.74738500050518396728503380630, −1.27894813709461865538816443306, −0.925731786783824312826518068671, −0.37862867084687999583677535292,
0.37862867084687999583677535292, 0.925731786783824312826518068671, 1.27894813709461865538816443306, 1.74738500050518396728503380630, 2.62350987118371064708438611967, 2.67885359059608155833061523795, 2.90078506726802688972652095096, 3.47198356819680165540182221223, 3.79220911587343167629823016699, 4.03260445489876216248547779420, 4.40699590305101401971494272686, 4.47819429893413615539785673564, 4.61827144021653111063085260929, 5.10687973454071378717316876648, 5.70963941476319391961838214009, 5.78186015180769933607696304184, 5.84885051643634919623635173668, 6.34499654029659436674064982108, 6.55797005893341862109985690471, 6.77708597494169768352291693761, 7.30798959440475872406608845753, 7.41884043844711695175031559377, 7.63854646362075158856285490530, 7.68187702819058762865372134318, 8.016802058409283456141586953644
